The Quantum to Classical Transition in Inflationary … · The Quantum to Classical Transition in...

The Quantum to Classical Transition inInflationary Cosmology

C. D. McCoy

Department of PhilosophyUniversity of California San Diego

Foundations of PhysicsMunich, 31 July 2013

Questions to Address

1. How does the measurement problem arise in cosmologicalstructure formation?

2. Which proposed solutions of the measurement problemare applicable to structure formation?

3. How does the Everett interpretation fare in explaining this“quantum to classical" transition?

A History of the Universe

The Cosmic Microwave Background Radiation

Θ(n) ≡ δT (n)

T0=∑`,m

a`mY`m(n)

The Power Spectrum of the Cosmic MicrowaveBackground

C` =1

2`+ 1

∑m

〈a∗`m,a`m〉 =2π

∫k2dk PΦ(k) ∆2

` (k)

Primordial Perturbations

Assume that the primordial spectrum follows a power law:

P(k) = A kns−1

Then the CMB data suggest that the spectrum is...1. approximately scale invariant (P(k) ≈ const);2. small amplitude (A << 1);3. and approximately Gaussian (ns ≈ 1);

Primordial Perturbations

Assume that the primordial spectrum follows a power law:

P(k) = A kns−1

Then the CMB data suggest that the spectrum is...1. approximately scale invariant (P(k) ≈ const);2. small amplitude (A << 1);3. and approximately Gaussian (ns ≈ 1);

...but a naive retrodiction of this spectrum (classical generalrelativity and relativistic hydrodynamics) results in all modesbeing generated beyond the horizon (acausal) at earlytimes/high densities.

Origin of the Primordial Spectrum


I Inexplicable acausal correlations—Horizon Problem!


I Inflation solves horizon problemI Inflation implemented by a scalar field with a certain

potential

Quantum Fluctuations as the Origin of the PrimordialSpectrum*

I Einstein-Hilbert Action and Action of a Scalar Field(Inflaton).

S = − 116π

∫d4x R

√−g +

∫d4x√−g[

12

gµν∂φ

∂xµ∂φ

∂xν− V (φ)

]I Linearize metric and scalar field, separating homogeneous

part from perturbation.

gµν(x , t) = gµν(t) + δgµν(x , t) φ(x , t) = φ(t) + δφ(x , t)

ds2 = a2(η)[− (1 + 2Φ) dη2 + (1− 2Ψ) γij dx i dx j ]

*Mukhanov, Feldman, and Brandenberger, Phys Rep 1992*Mukhanov, Physical Foundations of Cosmology 2005

Quantum Fluctuations as the Origin of the PrimordialSpectrum

I Introduce a gauge-invariant combination of the scalar fieldand metric perturbations, the Mukhanov-Sasaki variable v :

v(η, x) = a(δφ+φ′

HΨ) z = a

φ′

H

I And write the action in terms of v and z:

S2 =

∫Ldη d3x =

12

∫ (v ′2 − v,iv ,i +

z ′′

zv2)

dη d3x

v ′′k + (k2 − z ′′

z)vk = 0

Quantizing the Inflaton

I Define the momentum π canonically conjugate to v asπ = v ′.

I Promote v and π to operators obeying the standardcommutation relations.

I Decompose solutions into adiabatic modes with creationand annihilation operators and define the vacuum asak |0〉 = 0.

I Compute the power spectrum:

PΦ(k) =|Φk |2

2π2 =1

2π2 〈Φ(x , η)Φ∗(y , η)〉

Quantizing the Inflaton

I Define the momentum π canonically conjugate to v asπ = v ′.

I Promote v and π to operators obeying the standardcommutation relations.

I Decompose solutions into adiabatic modes with creationand annihilation operators and define the vacuum asak |0〉 = 0.

I Compute the power spectrum:

PΦ(k) =|Φk |2

2π2 =1

2π2 〈Φ(x , η)Φ∗(y , η)〉

I Quantum correlation function to ensemble average?〈0|Φ(x , η)Φ(y , η)|0〉 → 〈Φ(x , η)Φ∗(y , η)〉

Quantum Theory without Observers

I Quantum theory applied to cosmology must be a quantumtheory without observers.



1. Collapse TheoriesMartin, Vennin, and Peter, Phys. Rev. D 2012;Lochan, Das, and Bassi, Phys. Rev. D 2012;Cañate, Pearle, and Sudarsky, Phys, Rev. D 2013




2. Bohmian TheoriesValentini, Phys. Rev. D 2010;Pinto-Neto, Santos, and Struyve, Phys. Rev. D 2012




2. Bohmian TheoriesValentini, Phys. Rev. D 2010;Pinto-Neto, Santos, and Struyve, Phys. Rev. D 2012

3. Everett interpretation???

Classicality Condition for Many Worlds (Wallace)

I A history is a sequence of time-indexed PVMs.I The set of such histories generated from some such

sequence of PVMs is a history space P.I The history operator Cα of the history α is

Cα = α(n) . . . α(0).

I The decoherence functional is a complex function on pairsof histories (relative to a choice of state |ψ〉) given by

D(α, β) = 〈ψ|C†αCβ|ψ〉.

I A history space is said to satisfy the decoherencecondition or to be decoherent if the decoherence functionalbetween any two incompatible histories is zero.

I If P satisfies the decoherence condition, it is acoarse-graining of a (decoherent) history space which hasa branching structure relative to |ψ〉.

Decoherence in the Early Universe

I Decoherence is essentially the suppression of interferencewith respect to a system’s macroscopic degrees offreedom via interactions with microscopic degrees offreedom (internal or external “environment").



I Decoherence does not solve the measurement problem,but does it place a constraint on quantum theories ofstructure formation?




I In the literature on the quantum to classical transitionduring cosmological structure formation, decoherence hasbeen shown to occur in many ways: external scalar fields,short wavelength modes, non-linear coupling betweenmodes, spatial entanglement, thermal fluctuations afterreheating...




I In the literature on the quantum to classical transitionduring cosmological structure formation, decoherence hasbeen shown to occur in many ways: external scalar fields,short wavelength modes, non-linear coupling betweenmodes, spatial entanglement, thermal fluctuations afterreheating...

I What attitude should one take to these demonstrations?

Attitudes Toward Structure Formation and the EverettInterpretation

1. The ease of generating decoherence in the inflationaryscenario is further evidence that decoherence isubiquitous, thus by requiring decoherence the Everettinterpretation places a minimal constraint on theories ofstructure formation.

I What accounts for the ubiquity of decoherence andtherefore the direction of branching?



I What accounts for the ubiquity of decoherence andtherefore the direction of branching? Initial Conditions?




2. The variety of ways to decohere fields in the early universethat people have proposed raises theoretical worries aboutinflation or the Everett interpretation.

I Too little known about the inflaton and its interactions withother physical fields

I There is no “principled division" of the universe into systemand environment.





I Too little known about the inflaton and its interactions withother physical fields. Effective theory?

I There is no “principled division" of the universe into systemand environment.





I Too little known about the inflaton and its interactions withother physical fields. Effective theory?

I There is no “principled division" of the universe into systemand environment. Need to show consistency!

Conclusions

1. The standard theory of structure formation is a quantumtheory with the appearance of classical outcomes—thusthe quantum measurement problem appears in cosmologyas it does elsewhere in physics.

2. A solution of the measurement problem must be a quantumtheory without observers to be applicable to cosmology.

3. The Everett interpretation (modulo concerns overprobability) is applicable to cosmology, but is conservativeand unlikely to raise issues in this application—still, work isrequired to demonstrate satisfaction of the decoherencecondition.

4. Further work on the quantum measurement problem is ofrelatively little importance to cosmology; of far moreimportance is understanding the initial state of the universeand the nature of the inflaton.

Structure Formation and the Measurement Problem

Mukhanov, Physical Foundations of Cosmology

When we look at the sky we see the galaxies incertain positions. If these galaxies originated frominitial quantum fluctuations, a natural question arises:how does a galaxy, e.g. Andromeda, find itself at aparticular place if the initial vacuum state wastranslational-invariant with no preferred position inspace? Quantum mechanical unitary evolution doesnot destroy translational invariance and hence theanswer to this question must lie in the transition fromquantum fluctuations to classical inhomogeneities.


Decoherence is a necessary condition for theemergence of classical inhomogeneities and caneasily be justified for amplified cosmologicalperturbations. However, decoherence is not sufficientto explain the breaking of translational invariance. Itcan be shown that as a result of unitary evolution weobtain a state which is a superposition of manymacroscopically different states, each correspondingto a particular realization of galaxy distribution.


Most of these realizations have the same statisticalproperties. Such a state is a close cosmic analog ofthe “Schrödinger cat." Therefore, to pick an observedmacroscopic state from the superposition we have toappeal either to Bohr’s reduction postulate or toEverett’s many-worlds interpretation of quantummechanics.

Quantum Measurement Problem Today

I The quantum measurement problem is not apsuedo-problem.

I It is a conceptual stumbling block to the universalapplication of quantum theory, e.g. in cosmology.




I The quantum measurement problem is not solved bydecoherence.

I Decoherence does not destroy superpositions.





I Decoherence does not destroy superpositions.I The quantum measurement problem is not solved.

I Conceptual problems are not “solved". There are manyapproaches, some of which are coherent, of interest, etc.Others are not.





I Decoherence does not destroy superpositions.I The quantum measurement problem is not solved.

I Conceptual problems are not “solved". There are manyapproaches, some of which are coherent, of interest, etc.Others are not.

I The quantum measurement problem is not a severedifficulty.

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